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There is a huge amount of work on different kinds of theta functions, the theta correspondence, cohomology classes coming from special Schwartz classes via theta distribution, and much more. The aim of this text is to try to find joint construction principles while often leaving aside relevant but cumbersome details. The next steps after this prehistoric Part I will be directed to a description of the Howe operators introduced by Kudla and Millson and their special Schwartz forms and classes. This has as attractor the fact that the modular and automorphic forms arising naturally in context with these classes find very nice geometric interpretations of their Fourier coefficients and thus lead to an intriguing intertwining of elements of representation theory with algebraic and arithmetic geometry. The presentation here is in the spirit of my book on representations of linear groups. Though it may be seen as just another chapter, it has it has its own raison d’être and

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This survey is the expanded version of my talk at the AMS meet- ing in April 1997. I explain to non-experts how to use the orbit method, discuss its strong and weak points and advertise some open problems.

Eine gleichermassen aktuelle wie zusammenfassende Darstellung der wichtigsten Methoden zur Untersuchung der klassischen Gruppen fehlte bislang in deutschsprachigen Lehrbuchern. Indem der Autor die klassischen Gruppen sowohl aus algebraisch-geometrischer Sicht, wie auch mit Lieschen (infinitesimalen) Methoden studiert, schliesst er diese Lucke. Die von Grund auf behandelte Darstellungstheorie mundet im algebraischen Teil in der Brauer-Weylschen Methode der Zerlegung von Tensorpotenzen durch Youngsche Symmetrieoperatoren in irreduzible Teilraume. Auf der Ebene der Lie-Algebren wird die Klassifikation der irreduziblen Darstellungen durch hochste Gewichte durchgefuhrt. Besonderer Wert liegt auf einer ausfuhrlichen Erlauterung des Zusammenspiels der Gruppen und ihrer Lie-Algebren, die das Kernstuck der Lieschen Theorie ausmachen. In dieser Hinsicht dient das Buch auch als Einfuhrung in die Theorie der Lie-Gruppen; zur Parametrisierung wird dabei ausschliesslich die Matrix-Exponentialabbildung verwandt, wodurch ganz auf den aufwendigen Apparat der differenzierbaren Mannigfaltigkeiten verzichtet werden kann. Eine Fulle von Beispielen und Ubungsaufgaben dienen zur Vertiefung des Gelernten. Inhaltlich schliesst der Text unmittelbar an die Grundvorlesungen uber Analysis und Lineare Algebra an.

The area of automorphic representations is a natural continuation of studies in the 19th and 20th centuries on number theory and modular forms. A guiding principle is a reciprocity law relating infinite dimensional automorphic representations with finite dimensional Galois representations. Simple relations on the Galois side reflect deep relations on the automorphic side, called "liftings." This in-depth book concentrates on an initial example of the lifting, from a rank 2 symplectic group PGSp(2) to PGL(4), reflecting the natural embedding of Sp(2,≤) in SL(4, ≤). It develops the technique of comparing twisted and stabilized trace formulae. It gives a detailed classification of the automorphic and admissible representation of the rank two symplectic PGSp(2) by means of a definition of packets and quasi-packets, using character relations and trace formulae identities. It also shows multiplicity one and rigidity theorems for the discrete spectrum. Applications include the study of the decomposition of the cohomology of an associated Shimura variety, thereby linking Galois representations to geometric automorphic representations. To put these results in a general context, the book concludes with a technical introduction to Langlands’ program in the area of automorphic representations. It includes a proof of known cases of Artin’s conjecture. © 2005 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.

A uniform formulation, applying to all classical groups simultaneously, of the First Fundamental Theory of Classical Invariant Theory is given in terms of the Weyl algebra. The formulation also allows skew-symmetric as well as symmetric variables. Examples illustrate the scope of this formulation.

„Die Theorie der Zahlkörper ist wie ein Bauwerk von wunderbarer Schönheit und Harmonie; als der am reichsten ausgestattete Teil dieses Bauwerkes erscheint mir die Theorie der Abelschen und relativ-Abelschen Körper, die uns Kummer durch seine Arbeiten über die höheren Reziprozitätsgesetze und Kronecker durch seine Untersuchungen über die komplexe Multiplikation der elliptischen Funktionen erschlossen haben. Die tiefen Einblicke, welche die Arbeiten dieser beiden Mathematiker in die genannte Theorie gewähren, zeigen uns zugleich, daß in diesem Wissensgebiete eine Fülle der kostbarsten Schätze noch verborgen liegt, winkend als reicher Lohn dem Forscher, der den Wert solcher Schätze kennt und die Kunst, sie zu gewinnen, mit Liebe betreibt.“

We define theta functions attached to indefinite quadratic forms over real number fields and prove that these theta functions are Hilbert modular forms by regarding them as specializations of symplectic theta functions. The eighth root of unity which arises under modular transformations is determined explicitly.

As with the Neumann dynamical system, our purpose now is to introduce a dynamical system interesting in its own right, and then to show that it can, in some cases, be integrated explicitly by the theory of hyperelliptic Jacobians. More precisely, we can, following the ideas in the previous section, define an embedding of Jac C in an infinite dimensional space: Open image in new window On R1, we consider a simple class of vector fields X: those which assign to f a tangent vector in \(
X_f \in T_{R_1 ,f} \cong R_1
\) given by $$
X_f = P\left( {f,\dot f, \cdots ,f^{\left( n \right)} } \right),{\text{ P a polynomial}}
$$. Integrating this vector field means finding an analytic function f(x,y) s.t. $$
\frac{{\partial f}}
{{\partial y}} = P\left( {f, \frac{{\partial f}}
{{\partial x}},{\text{ }} \cdots {\text{, }}\frac{{\partial ^n f}}
{{\partial x^n }}} \right).
$$ By the Cauchy-Kowalevski Theorem, for all f(x,0) analytic in |x| < ε, there exists f(x,y) analytic in |x|,|y| < η solving this. What we want to do is to set up a sequence X1,X2,...of such vector fields called the Kortweg-de Vries hierarchy which a) commute [Xi,Xj] =0 — we must define this carefully — and b) are Hamiltonian in a certain formal sense, such that c) for all g, and for all hyperelliptic curves C of genus g: $$
\operatorname{Im} \left( {Jac - \Theta } \right) = \left[ {\begin{array}{*{20}c}
{orbit of all flows {\rm X}_n ,} \\
{i.e., all {\rm X}_n are tangent to image} \\
{and a codimension g subspace of} \\
{\sum {c_n {\rm X}_n } are even 0 on Image} \\
\end{array} } \right]

I Lie Groups and Lie Algebras.- II Elementary Representation Theory.- III Representative Functions.- IV The Maximal Torus of a Compact Lie Group.- V Root Systems.- VI Irreducible Characters and Weights.- Symbol Index.

In this chapter we assume that the reader is acquainted with the ordinary ideal theory in number fields. Cf. for instance [B7]. The first two sections should be read as technical background for Chapter 10, §2. On the other hand, although we strive for some completeness, once the reader sees the first results that the proper o-lattices form a multiplicative group, he can wait to read the other results until he needs them, as they are slightly technical. They are all classical, known to Dedekind, except possibly for the fact that a proper o-lattice is locally principal, which seems to have been first pointed out by Ihara [26]. The localization technique will be used heavily for the idelic formulation of the complex multiplication, as in Shimura [B12].

CONTENTSIntroduction § 1. Induced representations § 2. Representations of Lie algebras and infinitesimal group rings § 3. A special nilpotent group N § 4. Nilpotent Lie groups with one-dimensional centre § 5. Description of the representations of nilpotent Lie groups § 6. Orbits and representations § 7. Representations of the group ring § 8. Some generalizations and unsolved problems § 9. ExamplesReferences

ture quelconque (s,t) , soit LcRn un r~seau sur lequel q(x) prend des valeurs enti~res et soit p(x) : R n ~ C une fonction avec les propri~t~s suivantes : *) La fonction f(x) = p(x) e -2~q(x) , ainsi que D(x)f(x) et R(x)f(x) pour toute d~rivation D(x) d'ordre ~ 2 et tout polynSme R(x) de degr~ ~ 2 sont d~finies et appartiennent L2(R n) n Ll(e n) (E-X)p(x) = ~ p(x) , pour un entier ~ , ou E est l'op~rateur d'Euler et ~ le laplacien associ~ ~ q(x) . Alors la s~rie th~ta

A uniform formulation, applying to all classical groups simultaneously, of the First Fundamental Theory of Classical Invariant Theory is given in terms of the Weyl algebra. The formulation also allows skew-symmetric as well as symmetric variables. Examples illustrate the scope of this formulation.